«I have nothing against homosexuals, or any other group, promoting their agenda through normal democratic means.»

«I have nothing against homosexuals', or any other group's, promotion of their agenda through normal democratic means.»

My gut feeling is that the possessive with apostrophe must be used in the case of the ordinary noun «promotion» ; whereas the sentence containing the gerund «promoting» is better off without it. Moreover, I find Justice Scala's use of «their agenda» in the subordinate clause problematic (cf «I have nothing against homosexuals promoting their agenda through normal democratic means» and «I have nothing against any other group promoting its agenda through normal democratic means» and the question of whether the object of the gerund or proposition should properly be «agenda» or «agendas»), but this is a can of worms perhaps better left unopened....

Among other things, this is what Fowler says about the fused participle:

1.Women having the vote share political power with men.
2.Women's having the vote reduces men's political power.
3.Women having the vote reduces men's political power.

In the first, the subject of the sentence is women, and having (the vote is a true participle attached to women. In the second, the subject is the verbal noun or gerund having (the vote), and women's is a possessive case (.e. an adjective) attached to that noun. The grammar in these two is normal. In the third, the subject is neither women (since reduces is singular), not having (for if so, women would be left in the air without grammatical construction), but a compound notion formed by fusion of the noun women with the participle having. Participles so constructed, then, are called fused participles, as opposed to the true participle of No. 1 and the gerund of No.2.